Suppose 𝑆, 𝑇 ∈ ℒ(𝑉) are such that range 𝑆 ⊆ null 𝑇. Prove that (𝑆𝑇)2 = 0.

Given that \( S, T \in \mathcal{L}(V) \), where \( V \) is a vector space, and that the range of \( S \subseteq \text{null}(T) \), we need to prove that \( (ST)^2 = 0 \), i.e., \( STST = 0 \).

### Step-by-step proof:

#### Step 1: Understand the given condition
- \( S \) and \( T \) are linear operators on the vector space \( V \), meaning \( S, T: V \to V \).
- The range of \( S \), denoted \( \text{range}(S) \), is contained within the null space of \( T \), i.e.,
  \[
  \text{range}(S) \subseteq \text{null}(T),
  \]
  which means for every vector \( v \in V \), if \( v = S(w) \) for some \( w \in V \), then \( T(S(w)) = 0 \). In other words:
  \[
  T(S(w)) = 0 \text{ for all } w \in V.
  \]

#### Step 2: Apply the given condition to the composition of operators
Let \( v \in V \) be an arbitrary vector. We need to show that \( (ST)(ST)(v) = 0 \). To do so, first compute \( ST(v) \) and then apply \( ST \) again.

1. **First application of \( ST \):**
   - Apply \( T \) to \( v \), which results in \( T(v) \in V \).
   - Then apply \( S \) to \( T(v) \), so we get:
     \[
     ST(v) = S(T(v)).
     \]
   This vector, \( S(T(v)) \), belongs to \( \text{range}(S) \), and by the given condition, we know that:
   \[
   T(S(T(v))) = 0.
   \]
   
2. **Second application of \( ST \):**
   - Now apply \( ST \) again to \( ST(v) \). That is, apply \( ST \) to \( S(T(v)) \):
     \[
     ST(ST(v)) = S(T(S(T(v)))).
     \]
   Since \( T(S(T(v))) = 0 \), this simplifies to:
   \[
   S(0) = 0.
   \]

Thus, for any \( v \in V \), we have \( ST(ST(v)) = 0 \). Therefore, \( (ST)^2 = 0 \), or equivalently \( STST = 0 \), as required.

### Conclusion:
We have shown that \( (ST)^2 = 0 \) by leveraging the fact that \( \text{range}(S) \subseteq \text{null}(T) \). This completes the proof.

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Would you like further clarification or more details on any part of the proof?

Here are some related questions to explore:
1. What is the significance of the range and null space in linear transformations?
2. How would the proof change if the condition were \( \text{null}(S) \subseteq \text{range}(T) \)?
3. Can we generalize this result for higher powers, such as \( (ST)^k = 0 \) for \( k > 2 \)?
4. What are some examples of linear transformations where this condition holds?
5. How does this relate to the concept of nilpotent operators?

**Tip:** In linear algebra, understanding how the range and null space of operators interact can often reveal important structural properties about the operators, such as being nilpotent.

Suppose 𝑆, 𝑇 ∈ ℒ(𝑉) are such that range 𝑆 ⊆ null 𝑇. Prove that (𝑆𝑇)² = 0.

Explore a detailed step-by-step proof that (ST)² = 0 for linear operators S and T when the range of S is a subset of the null space of T. This problem highlights key concepts in linear algebra, including linear transformations, range, null space, and nilpotent operators. Suitable for college-level students studying linear algebra.

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